![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | β’ π· = (ODualβπ) |
oduval.l | β’ β€ = (leβπ) |
Ref | Expression |
---|---|
oduleval | β’ β‘ β€ = (leβπ·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6915 | . . . . 5 β’ (leβπ) β V | |
2 | 1 | cnvex 7941 | . . . 4 β’ β‘(leβπ) β V |
3 | pleid 17357 | . . . . 5 β’ le = Slot (leβndx) | |
4 | 3 | setsid 17186 | . . . 4 β’ ((π β V β§ β‘(leβπ) β V) β β‘(leβπ) = (leβ(π sSet β¨(leβndx), β‘(leβπ)β©))) |
5 | 2, 4 | mpan2 689 | . . 3 β’ (π β V β β‘(leβπ) = (leβ(π sSet β¨(leβndx), β‘(leβπ)β©))) |
6 | 3 | str0 17167 | . . . 4 β’ β = (leββ ) |
7 | fvprc 6894 | . . . . . 6 β’ (Β¬ π β V β (leβπ) = β ) | |
8 | 7 | cnveqd 5882 | . . . . 5 β’ (Β¬ π β V β β‘(leβπ) = β‘β ) |
9 | cnv0 6150 | . . . . 5 β’ β‘β = β | |
10 | 8, 9 | eqtrdi 2784 | . . . 4 β’ (Β¬ π β V β β‘(leβπ) = β ) |
11 | reldmsets 17143 | . . . . . 6 β’ Rel dom sSet | |
12 | 11 | ovprc1 7465 | . . . . 5 β’ (Β¬ π β V β (π sSet β¨(leβndx), β‘(leβπ)β©) = β ) |
13 | 12 | fveq2d 6906 | . . . 4 β’ (Β¬ π β V β (leβ(π sSet β¨(leβndx), β‘(leβπ)β©)) = (leββ )) |
14 | 6, 10, 13 | 3eqtr4a 2794 | . . 3 β’ (Β¬ π β V β β‘(leβπ) = (leβ(π sSet β¨(leβndx), β‘(leβπ)β©))) |
15 | 5, 14 | pm2.61i 182 | . 2 β’ β‘(leβπ) = (leβ(π sSet β¨(leβndx), β‘(leβπ)β©)) |
16 | oduval.l | . . 3 β’ β€ = (leβπ) | |
17 | 16 | cnveqi 5881 | . 2 β’ β‘ β€ = β‘(leβπ) |
18 | oduval.d | . . . 4 β’ π· = (ODualβπ) | |
19 | eqid 2728 | . . . 4 β’ (leβπ) = (leβπ) | |
20 | 18, 19 | oduval 18289 | . . 3 β’ π· = (π sSet β¨(leβndx), β‘(leβπ)β©) |
21 | 20 | fveq2i 6905 | . 2 β’ (leβπ·) = (leβ(π sSet β¨(leβndx), β‘(leβπ)β©)) |
22 | 15, 17, 21 | 3eqtr4i 2766 | 1 β’ β‘ β€ = (leβπ·) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3473 β c0 4326 β¨cop 4638 β‘ccnv 5681 βcfv 6553 (class class class)co 7426 sSet csts 17141 ndxcnx 17171 lecple 17249 ODualcodu 18287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-ltxr 11293 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-dec 12718 df-sets 17142 df-slot 17160 df-ndx 17172 df-ple 17262 df-odu 18288 |
This theorem is referenced by: oduleg 18291 odupos 18329 oduposb 18330 odulub 18408 oduglb 18410 posglbdg 18416 oduprs 32720 odutos 32724 mgccnv 32755 ordtcnvNEW 33562 ordtrest2NEW 33565 glbprlem 48080 |
Copyright terms: Public domain | W3C validator |